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Some new families of integral graphs

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Abstract

Let G i , i = 1,2,3 be finite simple graphs with vertex sets \(V\left( {{G_1}} \right)\, = \,\left\{ {{u_1},{u_2},...,{u_{{n_1}}}} \right\},\,V\left( {{G_2}} \right)\, = \,\left\{ {{v_1},{v_2},...,{v_{{n_2}}}} \right\}\;and\;\,V\left( {{G_3}} \right)\, = \,\left\{ {{w_1},{w_2},...,{w_{{n_3}}}} \right\}\;\). In this paper, for each ordered triple of graphs (G 1, G 2, G 3), we define a new composition ψ(G 1, G 2, G 3) : The vertex set of ψ(G 1, G 2, G 3) is (V(G 1) ⨄ V(G 2)) × V(G 3) and adjacency between vertices of ψ(G 1, G 2, G 3) is defined by:

  1. 1.

    (u i , w l ) adj (v k , w l ) for each i = 1, 2,..., n 1 and k = 1, 2,..., n 2 and l = 1, 2,..., n 3.

  2. 2.

    (v i , w j ) adj (v k , w l ) if and only if

    1. i)

      i = k and w j adj w l , in G 3 or

    2. ii)

      j = l and v i adj v k in G 2

  3. 3.

    (u i , w j ) adj (u k , w j ) whenever u i adj u k in G 1, for each j = 1, 2,..., n 3.

We obtain the Adjacency spectrum, Laplacian spectrum and Q-spectrum of ψ(G 1, G 2, G 3). As an application, many new infinite families of R-integral graphs where R ∈ {A, L, Q} are constructed.

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Authors and Affiliations

  1. Department of Mathematics, St. Aloysius College, Edathua, Alappuzha, 689573, India

    G. Indulal

  2. Department of Mathematics, Bharathidasan University, Tiruchirappalli, 620024, India

    R. Balakrishnan & A. Anuradha

Authors
  1. G. Indulal
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  2. R. Balakrishnan
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  3. A. Anuradha
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Correspondence to G. Indulal.

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Indulal, G., Balakrishnan, R. & Anuradha, A. Some new families of integral graphs. Indian J Pure Appl Math 45, 805–817 (2014). https://doi.org/10.1007/s13226-014-0090-7

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  • DOI: https://doi.org/10.1007/s13226-014-0090-7

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